Exact T-count of the line family

Determine whether the cubic phase family \(f_n=\sum_i x_i x_{i+1}x_{i+2}\) has exact T-count \(2n+1\) for every \(n\).

Background

The isotropy floor gives $2n+1$ for the line family for $n\ge5$, and explicit or computationally verified constructions attain this value through $n=12$.

The paper does not establish equality for all lengths, so the general exactness of this family remains unresolved.